Goldrei's Classic Set Theory For Guided Independent Study. I don't necessarily think he's the greatest expositor, but his educational philosophy is spot on. For instance, he starts with the real number system and asks: how do we know this system exists? One possible answer is: because the set of all Dedekind cuts of rational numbers can be made into a real number system in a natural way. Okay but how do we know a rational number system exists? Easy: we can build rational numbers as equivalence classes of integers. But wait! Perhaps there is no integer number system. But that can't be, because we can build integers out of naturals. Okay, but maybe there doesn't exist a natural number system.
At this point, the reader has an epiphany. The existence of all the major number systems can be demonstrated using set theory alone - that is, if we can build a natural number system. So if we can build such a system, then WOW! Set theory is POWERFUL. Its only at this late stage in the game that Goldrei actually starts talking about the ZFC axioms. And it works great!
Too many math books start with axioms, or esoteric definitions, without giving the reader any intuition about why they should care. Goldrei's book is a breath of fresh air in this regard. Truly, a remarkable book.
If you're concerned about time, I don't think reading Calculus by Spivak is the best thing to do.
Either Zorich or Apostol is a great choice. I would say that they're "intermediate" in difficulty. Zorich contains more in-depth discussions of topics, and more examples than does Apostol.
If your goal is only to move on to Royden, you'll probably cover the material more quickly in Apostol. Zorich covers a number of topics not addressed in Apostol, such as vector analysis and submanifolds of $\mathbf{R}^n$. These are important topics, but not direct prerequisites for Royden. Still, I think with all the Lagrange multipliers and similar tools people use in economics, the submanifold topic is important if you want to understand the theory very clearly.
Zorich's first volume is quite concrete, whereas Apostol becomes abstract more quickly. This is probably because he doesn't want to duplicate what would be in a rigorous calculus book like his Calculus, although he does this more than Rudin's book does. His analysis book was for second- or third-year North American students, whereas Zorich's is, at the outset, for first-year Russian ones. Russian students have typically had some calculus in high school, but the practical portion of learning calculus continues into their first-year of university, with harder problems. So in Zorich I, you deal with hard problems on real numbers, rather than delving straight into metric spaces as you would in Apsotol's book.
Zorich covers only Riemann integration, whereas Apostol has chapters on Riemann-Stieltjes integration in one variable, Lebesgue integrals on the line, multiple Riemann integrals, and multiple Lebesgue integrals. The treatment of Lebesgue integration is less abstract than in more advanced books. Since it's limited to $\mathbf{R}$ or $\mathbf{R}^n$, it's more elementary, but at the same time there is some loss in clarity compared to the abstract theory on measure spaces. One reason to use Apostol might be a sort of introduction to the Lebesgue theory before returning to it at a higher level and "relearning" certain parts of it. Whether you'd want this is up to you.
The fact that both Rudin and Apostol have chapters on Riemann-Stieltjes, rather than Riemann, integration, indicates to me that they assumed students had already studied Riemann integrals rigorously, and would be ready for a generalized version right from the start. Considering the type of calculus courses most students take these days, this is rarely the case now. Zorich doesn't have this problem.
All in all, for a typical student who is good at math but didn't learn their calculus from a book like Spivak's or Apostol's Calculus, I think Zorich is the better choice because of the more concrete approach in the first volume (this doesn't necessarily mean easier). On the other hand, time constraints might cause you to prefer Apostol's analysis book.
EDIT: An important point that I neglected to mention is that Zorich's book will be much better than Apostol's if you aren't yet acquainted at all with multivariable calculus. A practical knowledge of some multivariable calculus is probably one of the tacit assumptions that Apostol and Rudin make about their readers, which is what allows them to deal with multivariable calculus in a briefer and more abstract way. Compare Apostol's 23-page chapter on multivariable differential calculus to Zorich's 132 (in the Russian version).
EDIT: Based on your later comments, I would suggest that reading
Spivak's Calculus,
Whichever you prefer of Apostol's Mathematical Analysis or Rudin's Principles of Mathematical Analysis.
would be a reasonable plan.
However, before beginning the multivariable calculus parts of those books, it would be best to learn some linear algebra and multivariable calculus from another source. This could be Volume 2 of Apostol's Calculus. You could instead skip straight to the multivariable part of Volume 1 of Zorich, but you'd have to learn the necessary linear algebra elsewhere first. I don't recommend Spivak's Calculus on Manifolds if you want to learn multivariable calculus for the first time. Also, you won't need Munkres - you'll get enough topology to start with in whichever other book you read.
EDIT: In answer to your additional question, these topics are mostly not discussed in Spivak.
However, Spivak is an excellent introduction to the mathematical way of thinking. That is, although you will not learn all the specific facts that arise in higher-level books (you do learn many, of course), you will learn to read and understand definitions, theorems and proofs the way mathematicians do, and to produce your own proofs. You will become intimately familiar with real numbers, sequences of real numbers, functions of a real variable and limits, so you will have examples in mind for the more general structures introduced in topology. You will also solve difficult problems.
So it is not that you will know topology already when you've read Spivak's book, it's mainly that it ought to be easier for you to learn because you will have improved your way of approaching mathematical questions. Countable sets are in fact discussed in the exercises to Spivak, however.
I can't guarantee that your trouble will "go away," but there is a good chance it will.
Also feel free to use Zorich rather than Rudin or Apostol, after Spivak, or even to jump straight to the multivariable part of Zorich at the end of Volume 1 and start reading from there.
Best Answer
We use Kostrikin's Introduction to Algebra here in China. We use Zorich for calculus too and I feel they have a somewhat similar style. Try it out, but I think the English version may be fairly old. The latest Russian edition is the 3rd edition.